login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A229723
Expansion of psi(q) * chi(-q^3) * phi(-q^6) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
3
1, 1, 0, 0, -1, 0, -2, -2, 0, -2, 2, 0, 0, 0, 0, 4, -1, 0, 0, 0, 0, 0, 2, 0, 2, 3, 0, 0, -2, 0, 0, -2, 0, -4, 0, 0, 2, 0, 0, 0, -2, 0, -4, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, -2, -4, 0, 0, 2, 0, 4, 0, 0, 4, -1, 0, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 0, -2, 0, -2
OFFSET
0,7
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion eta(q^2)^2 * eta(q^3) * eta(q^6) / (eta(q) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 1, -1, 0, -1, 1, -3, 1, -1, 0, -1, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 13824^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128583.
a(3*n + 2) = 0.
EXAMPLE
G.f. = 1 + q - q^4 - 2*q^6 - 2*q^7 - 2*q^9 + 2*q^10 + 4*q^15 - q^16 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ q^3, q^6] EllipticTheta[ 4, 0, q^6] EllipticTheta[ 2, 0, q^(1/2)] / (2 q^(1/8)), {q, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^2 QPochhammer[ q^3] QPochhammer[ q^6]/ (QPochhammer[ q] QPochhammer[ q^12]), {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^6 + A) / (eta(x + A) * eta(x^12 + A)), n))};
CROSSREFS
Cf. A128583.
Sequence in context: A214667 A214665 A352557 * A258040 A215879 A114700
KEYWORD
sign
AUTHOR
Michael Somos, Sep 27 2013
STATUS
approved